Kac-Moody群、解析正则性条件与城市
Kac-Moody groups, analytic regularity conditions and cities
摘要：
最小代数卡茨-穆迪群和孪生建筑物之间的关系是众所周知的，因为是在一个方向上的正式完成和仿射建筑物之间的关系。然而，当一个方向上的Kac-Moody群的完成破坏了相反的BN对时，就不再存在双子楼了。由于类似的原因，至今还没有任何建筑物可以用于仿射Kac-Moody群的解析完备化。在这篇文章中，我们构建了一种新型的孪生建筑，称为孪生城市，它与实数或复数上的仿射解析Kac-Moody群相关联。双子城由两组建筑组成。通过证明有限维微分几何经典结果的无限维版本，我们描述了城市在无限维微分几何中的应用：例如，我们证明了希尔伯特空间中的等参子流形中的点对应于城市中的所有腔室。在续集中，我们将描述正式完成的双城理论。
英文摘要：
The relationship between minimal algebraic Kac-Moody groups and twin buildings is well known as is the relationship between formal completions in one direction and affine buildings. Nevertheless, as the completion of a Kac-Moody group in one direction destroys the opposite BN-pair, there exists no longer a twin building. For similar reasons, there are so far no buildings at all for analytic completions of affine Kac-Moody groups. In this article we construct a new type of twin buildings, called twin cities, that are associated to affine analytic Kac-Moody groups over the real or complex numbers. Twin cities consist of two sets of buildings. We describe applications of cities in infinite dimensional differential geometry by proving infinite dimensional versions of classical results from finite dimensional differential geometry: For example, we show that points in an isoparametric submanifold in a Hilbert space correspond to all chambers in a city. In a sequel we will describe the theory of twin cities for formal completions.
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